This semester, I teach the master's course

Please check the course's website for more informations. Most important, you should not forget to indicate your preferences for each problem sheet. How this work in general is explained below. In the previous semesters, I prepared and gave the following courses at the University of Wrocław:

- An Introduction to String Theory (WS 22/23),
- Lie Algebras and Lie Groups (SS 22), and
- Quantum Field Theory (SS 22)

For completeness, as a PhD student, I also gave tutorial for

- Theoretical Mechanics, and
- String Theory I,

both taught by Dieter Lüst at the LMU Munich.

## Courses

Here is an overview about my courses. Listed are the most recent completed semesters. If you are interested in any other installments, please follow the links above.

### An Introduction to String Theory

USOSOne of the greatest puzzles in theoretical physics is to find a consistent quantum theory of gravity, the only one of the four fundamental forces we do not know how to quantise yet. Although decades of intensive work, we have no conclusive answer to this problem. Currently one of the main contender in race towards quantum gravity is String Theory. It is based on a simple but far reaching idea, to substitute point-like particles with extended objects, strings and membranes. In this course, we will explore the implications of this idea. More precisely, we study the bosonic string and see that it not onl describes gravity but also gauge theories on D-branes. Gauge theories are required to describe the remainig three fundamental forces. This renders string theory to a theory of everything. The course is tailored towards master and PhD students who are familiar with

- electrodynamics
- special relativity
- quantum field theory.

Basic knowledge of general relativity is an advantage, but you also should be find if you did not have attended the GR course yet. There are many good books on the subject, four that I like are

- Barton Zwiebach: A First Course in String
- Blumenhagen, Lüst, Theisen: Basic Concepts of String Theory
- Polchinski: String Theory, Volume 1
- Johnson: D-Branes

Moreover, David Tong has some excellent lecture notes. We will have 2.5 hours of lectures and 2.5 hours of tutorial each week. Exercises will be posted here a week before the tutorial they are discussed in. Please keep in mind that active participation in the tutorials is important to pass the course. M.Sc. Luca Scala will be the assistant for the tutorials. Please fell free to contact him or me if you have any questions.

There are many other interesting approaches to Quantum Gravity and string theory is by far not the only one. There is another course this semester, Introduction to Quantum Gravity, which I highly recommend.

**Important:** Students will be assigned to exercise problems by the system described below, before the tutorial. Please familiarise yourself with this system and do not forget to indicate your preferences.

Additional material for the individual lectures, including the exercises which we discuss in the tutorials, will appear here in due time.

- Why, what and how - from the relativistic particle to the Nambu-Goto actionTutorial13.03.2023 11:00, exercise
Please note that there are no exercise assignments for the first tutorial. However, we will use a few mintues to explain how exercise problems are assigned. To prepare, it would be great if you could already follow the instructions outlined in the pdf you find under the exercise link above. For the remainder of the tutorial, we continue with the lecture. This gives us time to motivate why we need string theory, to look at the relativistic point particle and to show how it can be generalised to the string.

*Note:*There were some questions today in the lecture about the history of String Theory. A nice account of this topic is given by the short article arXiv:1201.0981 of John Schwarz, one of the founders of the field. - Classical string and the Polyakov action
- Symmetries of the classical string, especially, Weyl invariance
- Mode expansion and open/closed string boundary conditions
- Old covariant quantisation, Virasoro algebra and physical states
- Physical spectrum, massless states, no-ghosts theorem, critical dimension $D=26$
- Light-cone quantisation (tutorial) and modern covaraint quantisation in the BRST formalism with bc-ghost system (lecture)
- Conformal field theory (CFT) and operator product expansion (OPE) in a nutshell
On CFT, one could and should give a full semester course. We cannot do this here, but I try to give you a self-contained introduction. CFT is crucial to discuss interactions between strings. Therefore, we have to gain a basic understanding of it.

- Vertex operators and a first glance at string perturbation theory
[BLT] section 6.1 and 6.2 String perturabation theory is very beautiful. It crucially depends on the moduli space of Riemann surfaces. Unfortunately, we do not have enough to look at this topic in more detail. For those of you how are interested, section 6.2 of the book by Blumenhagen, Lüst and Theisen is a very good starting point. To make you a bit more curious, here is a picutre of a tree-point vertex in the complex plane.

Much more details how it arises can be found in section 4.2 of my PhD thesis Hassler:2015pea.

- Low energy effective actions and compactifications
- Closed strings on circles and T-dualtiy
- D-branes and gauge theories

### Quantum Field Theory

USOSThis is the mandatory Quantum Field Theory course of the Master in Theoretical Physics at the University of Wrocław. It is tailored towards master and PhD students who are familiar with

- quantum mechanics
- electrodynamics
- special relativity
- quantum electrodynamics.

There are many good books on the subject, four that I like are

- [PeskSchr] Peskin, Schroeder: An introduction to quantum field theory
- [Ryder] Ryder: Quantum Field Theory
- Weinberg: The Quantum Theory of Fields, Volume 1 & 2
- Zee: Quantum Field Theory in a Nutshell.

We will follow mostly the first one in the lectures. There will be 2 hours of lectures and 2 hours of tutorial each week. Exercises will be posted here a week before the tutorial they are discussed in. Please keep in mind that active participation in the tutorials is important to pass the course. For more information, please refer to the syllabus (in Polish) or contact me directly.

**Important:** Students will be assigned to exercise problems by the system described below at the Thursday, 9:00 pm, before the tutorial. Please indicate your preferences by then.

**Exam:** As we decided together, the exam will be a written exam. It will take place on Tuesday, the 28th of June 2022 at 10:00 am in room 511. *Note that we will start at this time! Please, be there a bit earlier to avoid problems.* To prepare for the exam, you should have a look at the practise exam, which we will also discuss in the last tutorial. It may give you a better idea about possible questions and is intended to help you prepare.

**Retake Exam:** The retake exam will take place at Friday, the 9th of September 2022 10:00 am in room 447 (the same room where we had the lectures).

Additional material for the individual lectures, including the exercises which we discuss in the tutorials, is given below:

- Reminder of spin 0 and $1/2$ fields
[PeskSchr] sections one and two Please note that there are no exercise assignments for the first seminar. Instead, we continue with the lecture.

- Reminder of spin 1 fields and abelian gauge symmetries
[Ryder] sections 3.3 and 4.4 - Non-abelian gauge symmetries and Lie groups
[PeskSchr] section 15 except for 15.3 or alternatively [Ryder] sections 3.5 and 3.6 - Path integral in quantum mechanics and for the scalar field
[PeskSchr] section 9.1 - Generating functional, interactions and Feynman rules
[PeskSchr] section 9.2 - Path integral for fermions, Grassmann numbers, chiral anomaly (in the exercise)
[PeskSchr] section 9.5 - Path integral for spin-1 bosons, ghost fields
[PeskSchr] section 9.4 - One loop effects in QED: field-strength renormalisation and self-energy
[PeskSchr] section 7.1 - Dimensional regularisation and superficial degree of divergence
Peskin&Schroeder sections 7.5 and 10.1 - One-loop renoramlised $\phi^4$ theory
[PeskSchr] section 10.2 - Renormalisation group flow
[PresSchr] section 12.1 **Note:**Last time in the exercise, we discussed the renormalisation of Yukawa theory. In particular, we computed all counterterms at one loop. We did not finish this task, and I got the impression that at least some of you were struggling with this task. But in this exercise, we will need the counterterms to compute the β-functions. Therefore, we will not have a new problem set this week but instead finish the work on the counterterms together and discuss how we use them to compute the one-loop β-functions for Yukawa theory.*Please still prepare as well as possible by revisiting our problem sheet from last week.* - The Callan-Symanzik equation and $\beta$-functions
[PresSchr] sections 12.2 As usual, we will assign students to the different subparts of this exercise. But it might be a good idea to try to solve it completely as a preparation for the exam. It contains most of the central concepts and techniques we learned in this course in a still tractable way.

- Gravity, quantum gravity, one-loop $\beta$-functions of a two-dimensional $\sigma$-model and string theory
- Spontaneous symmetry breaking and the Higgs mechanism
[PresSchr] sections 11.1 and 20.1 In the exercise, we discuss the practise exam. Also keep in mind that this is your last chance to hand in your bonus point problem about the one-loop β-functions of the non-linear σ-model (worth 6 additonal points).

- BRST symmetry, physical Hilbert space
[PeskSchr] section 16.2-16.4

### Lie Algebras and Groups

USOSLie algebras describe infinitesimal symmetries of physical systems. Therefore, they and their representation theory are extensively used in physics, most notably in quantum mechanics and particle physics. This course introduces semi-simple Lie algebras and the associated Lie groups for physicists. We discuss the essential tools, like the root and weight system, to efficiently work with them and their representations. As on explicit application of the mathematical framework, we discuss Grand Unified Theories (GUT). Moreover, we show how modern computer algebra tools like LieART can significantly help in all explicit computations throughout the course.

A simple example is the visualisation of the root system of $E_6$ projected on the Coxeter plane, which you can see here. If you want to understand how it is created and connected to particle physics, you should take this course. Basic knowledge of core concepts in linear algebra, like vector spaces, eigenvalues and eigenvectors, is assumed. Some good books about the topic are:

- Fuchs and Schweigert: Symmetries, Lie Algebras and Representations
- Gilmore: Lie Groups, Lie Algebras, and Some of Their Applications
- Fulton and Harris: Representation Theory
- Georgi: Lie Algebras In Particle Physics: from Isospin To Unified Theories

The article Phys. Rep. 79 (1981) 1 by Slansky and the manual of the LieART package are good references, too.

**Note:** At multiple occasions, we will use Mathematica and it might be a good idea to set it up on your computer. Following the instructions below you should be eventually able to run the

**Exam:** As we decided together, the exam will be a written exam. It will take place on Friday, the 1st of July 2022 at 10:00 am in room 511. *Note that we will start at this time! Please, be there a bit earlier to avoid problems.* To prepare for the exam, you should have a look at the practise exam. It may give you a better idea about possible questions and is intended to help you prepare. Take it serious and try to solve it in two hours.

**Retake Exam:** We now have a consensus for the date of the retake exam. It will take place at Wednesday, the 7th of September 2022 10:00 am in room 447 (the same room where we had the lectures).

- Introduction and motivationLecture28.02.2022 11:15
- Some mathematical preliminariesLecture07.03.2022 11:15
- Classical matrix groupsLecture14.03.2022 11:15
- Cartan subalgebraLecture21.03.2022 11:15
- Root systemLecture28.03.2022 10:15
- Simple root and Cartan matrixLecture04.04.2022 10:15
- Classification and Dynkin diagramsLecture11.04.2022 10:15
- Irreducible representationsLecture25.04.2022 10:15
- $\mathfrak{su}$($N$) representations and Young tableauxLecture02.05.2022 10:15
- Highest weight representationsLecture09.05.2022 10:15
- Characters and Weyl groupLecture16.05.2022 10:15
- Decomposition of tensor products and regular subalgebrasLecture23.05.2022 10:15
- Special subalgebras and branching rulesLecture30.05.2022 10:15
- Particle theory and the standard modelLecture05.06.2022 10:00
- The Georgi–Glashow modelLecture13.06.2022 10:15

## Generalities

### Mathematica

For computations we use Mathematica. It is a very powerful tool with unfortunately a quite high price for a license. Students can get a discount on licenses. If your budget is not sufficient for a student license, you can use the Wolfram Engine for Developers. After creating a Wolfram ID, it can be downloaded for free. The Wolfram Engine implements the Wolfram Language, which Mathematica is based on. But, it lacks the graphical notebook interface. Fortunately, some excellent free software called Jupyter Notebook fills the gap. Both can be connected with the help of Wolfram Language for Jupyter project on GitHub. Getting everything running might require a little bit of tinkering. But in the end, you get a very powerful computer algebra system for free. Finally, you should install LieART by following the "Manual Installation" instructions.

### Assignment of problems

Solving the exercise problems for a course is very important. It helps to practice the concepts and ideas introduced during the lecture, and you will also be graded for the solutions you present during the tutorials. But at the same time, one of the most annoying questions for the students and the lecturer is: "Who would like to present his solutions to the next problem?".

Therefore, we use the following system to assign students to problems based on their preferences:

- You need to log in with your USOS account (every student at the University of Wrocław should have one). To do so, click on the small closed door in the top left corner of this window and enter your credentials in the window which pops up. The first time you do this, you will be asked to give this website minimal access to your USOS profile.

- After a successful login, you can go to your course above and find the new link "manage" after each lecture with an exercise. If you do not see this link, check if you are logged in (the small closed door you clicked in the last step should be slightly open now). Second, verify that the course you are looking at is indeed your course. When the problem still persists, please get in touch with me.

- If you click "manage", you will find a list of all the problems that we discuss during the exercises. If problems have not yet been assigned to students, you can indicate your preferences by sorting this list. Entries on the top have the highest priority, while those at the bottom have the lowest. You sort them by drag&drop with either the mouse or your finger if you work with a touchscreen. Once this is done, do not forget to click the "Save" button at the bottom.

You can always come back later and revisit your choice or change it until the assignments are fixed. Once this happens, you will get an email and see the students' names to present the various problems. The backup candidate (the second name) should be ready to take over if required.

- Make sure you log out at the end of your session by clicking on the now slightly opened door you arleady used to login.

The assignments made by this system are binding. For every problem you present you can get up to three points. These points are added up and used at the end of the course to calculate your grade. If you cannot present an assigned problem, you will get zero points and put your backup on the spot. Therefore: **Please prepare properly and in case of any emergencies, let us know timely.** Backup candidates can earn extra points (up to 1.5) by presenting a problem. But they can also lose the same amount if they are not prepared.

Problems are assigned completely automatically after the criteria: Everybody in the course should present the same number of problems. The indicated preferences are taken into account. If several students have the same preferences, the one who submitted them the earliest wins. You do not have to submit any preferences at all. In this case, the system assumes that you do not care which problem you have to present.