Introduction to Higher-Spin Gauge Theory
This is a backup of the website for the (Kramers) course on higher-spin gauge theory at Utrecht U in Spring 2014.
Lecturer: Prof. dr. Mikhail Vasiliev (Lebedev Physical Institute, Moscow)
Teaching assistant: Jules Lamers
Symmetries play a fundamental role in physics. This course will give an introduction to field theories exhibiting various types of symmetries, including Yang-Mills theory, supersymmetric field theories, gravity, supergravity and higher-spin gauge theories. The latter are highly symmetric relativistic theories conjectured to underly quantum gravity and String Theory.
Topics covered will include the Wigner classification of elementary particles, differential forms and Cartan-MacDowell-Mansouri gravity, and star-product algebras. Implications of higher symmetries for the fundamental concepts of space-time geometry as well as relevant aspects of the AdS/CFT correspondence will be discussed.
Note: The lecture notes can be found here (version of February 5, 2015). We are working on an updated version, so any feedback is welcome!
On this page you can find more about
- the lectures, including the venue and topics,
- the lecture notes (updated weekly),
- the tutorials (including the presentations of starred problems),
- the exam and retake (including the precise material and other practicalities), and
- further information for students, including details about the examination.
Lectures (updated regularly)
There are lectures on Thursdays at 15:15–18:15 on March 6, 13, 20, 27; April 3, 10, 17, 24; May 1, 8, 15 (?), 22. The following more detailed schedule includes the venue, which is time dependent, so make sure to check this website regularly.
| Date | Venue | Topics |
| March 6 | MIN 012 | Introduction. Brief recap of (Lie) groups and Lie algebras. The Poincaré group. Scalar fields, spinor fields, and massive and massless vector fields as unitary representations of the Poincaré group. |
| March 13* | MIN 211 | Wigner classification of elementary particles as representations of the Poincaré group. Young diagrams. Overview of field equations for general relativistic fields. |
| March 20 | BBG 061 | Lagrangian formulation for totally symmetric traceless higher-spin fields (Fronsdal theory). Local symmetries. Yang-Mills theory. |
| March 27 | BBG 061 | Noether charges. Diffeomorphisms. Differential forms, with two applications in physics. Gravity in the Cartan formalism. |
| April 3 | BBG 061 | Cartan gravity continued: Minkowski vacuum. Field equations for pure gravity. Coupling to matter. Problem of higher-spin fields in Minkowski background. (Anti-)De Sitter space. |
| April 10 | Unnik 001 | (Anti-)De Sitter space continued. Clifford algebras. Supersymmetry. |
| April 17* | BBG 161 | Supersymmetry continued. N=1 Wess-Zumino model (scalar multiplet), in Majorana, Weyl, and tensor formulations. N=1 vector multiplet. Wigner analysis for super-Poincaré algebra. |
| April 24 | MIN 211 | Wigner analysis for super-Poincaré algebra continued. Poincaré supergravity. |
| May 1 | MIN 211 | Anti-De Sitter supergravity. Frame-like formulation for higher-spin fields. Conformal symmetry. |
| May 8 | MIN 205 | Higher-spin algebra as symmetries of massless scalar. Weyl algebra. Star product. Frame-like formulation in spinor notation in 4d. Note: The lecture is shifted to 9:45–13:00 due to the department day. |
| May 15 | No lecture; instead there is an extra tutorial. | |
| May 22 | MIN 208 | Note: The material covered in this lecture is not part of the exam. |
*) Unfortunately there is some overlap with the QFT retake (March 13) and the exams (April 17). Students are encouraged to come to the lecture when they are finished, read the lecture notes afterwards to fill in the gaps, and ask any questions.
In addition, Professor Vasiliev gives two seminars at our institute. Students are invited and encouraged to attend!
| Date | Venue | Topics |
| May 23 21 | MIN 401 | String seminar at 11:15–12:15, including a direct application of the machinery from the lecture of May 8. Note: Due to the PLANCKS symposium the seminar is shifted to Wednesday at the usual time. |
| May 28 | MIN 211 | Colloquium at 16:00–17:00 with some general aspects of and perspectives on higher-spin gauge theory. |
Lecture notes (updated regularly)
The lecture notes until now can be found here (version of February 5, 2015). This file will be updated a few days after each lecture. It contains three types of problems:
- Exercises ask you to check a statement that is made in the lecture notes, and should be quite straightforward.
- Problems are similar to exercises, but require some more thought and effort. Note: For your convenience some problems are now marked ‘optional’ (version of April 15).
- Starred problems are problems that may be presented by a student during the tutorials. These problems usually go into some details that are important but have been omitted from the lectures in view of time, and the presentation contributes yields up to one bonus point for the final grade.
Note: To avoid situations where two students are preparing the same starred problem, please come to Jules to claim (register for) such a problem. More about the presentations can be found below.
Tutorials (updated regularly)
For the organization of the tutorials we have asked your opinion via this form. The outcome is that tutorials will take place on Mondays at 13:15–15:00 in the seminar room, MIN 401. (Unfortunately there is no time slot that suits everyone, but this one works for almost everyone. The few students who cannot make it on Monday can come by to ask any questions on Thursdays before the lecture.)
Note: To make the tutorials more efficient, students are asked to try and solve the exercises and problems whilst going through the relevant sections of the lecture notes—see the schedule below—before the tutorials. Students who manage to solve all the exercises and problems are still advised to come to the tutorial to check their solutions.
| Date | Sections | Remarks about the exercises/problems |
| March 24 | 1–2 | E2.1, P2.1, E2.10, P2.5 and P2.6 (now E2.11) updated (version of March 20). If you are done you may go on to Sect 3.1, but skip P3.2 for now. |
| March 31 | 3.1–3.3 | Skip the former P3.2 (end of Sect 3.1.2). P3.3 (now P3.2) updated, P3.8 upgraded to P*3.1, P3.10 downgraded to E3.6 (version of March 28). Interested students are invited to go through Section 3.4 as well. E3.4 and section 3.2.3 updated (version of April 3). |
| April 7 | 4–5 | P5.5 updated (version of April 4). E4.4, E5.4 and P5.1 updated (version of April 8). |
| April 14 | 6–7 | P6.1, P6.2, P6.4, E6.2, P6.6, P7.1 and P7.7 updated (version of April 11). |
| April 21 22 | catch up | P4.4, P5.1, P5.4 and P6.8 updated; P5.5, P6.8 and P7.6 marked as optional (version of April 15). Due to Easter the tutorial is shifted to Tuesday at 14:00–16:00. There are no new exercises, so use your time to catch up! |
| April 28 | 8 | E3.6 updated (version of April 23). |
| May 5 6 | 9–10, 11.1, 11.4 |
Problems 10.1, 10.3, 10.4, 10.6–10.9 and 11.15 optional (version of May 5). Due to Easter the tutorial is shifted to Tuesday at 14:00–16:00. |
| May 12 | 11–13 | |
| May 15 | Extra tutorial at 15:15–17:00. | |
| May 19 | 14–15.3 | |
| May 23 22 | Extra tutorial at 13:00–15:00. Note: Due to the PLANCKS symposium the tutorial is shifted; if the new time is not convenient you may come to our offices to ask questions on Friday (13:15–15:00). | |
| May 26 | Exam | The exam is at 13:30–16:30 in MIN 202. For more information see below. |
| May 28 | Last presentations of starred problems at 11:30, in the ITP library. | |
| July 7 | Retake | The retake is at 14:00–17:00 in MIN 401 204. For more information see below. |
Presentations: After registration for a starred problem the student has two weeks to solve the problem and prepare a presentation. The presentation may be use blackboard or slides; in any case the student should be able to answer any questions on the blackboard. There is a strict upper bound of 25 minutes (plus 5 minutes for questions), but the presentation may of course be shorter.
Note: The last opportunity for the students to present a starred problem is on Wednesday May 28 (time to be determined together with the students).
The claimed starred problems can be found in the following schedule for the presentations.
| Date | Problem* | Name |
| April 14 | 3.3 (reduction on S1) | Govert Nijs |
| April 21 22 | 3.2 (Wigner and massless fermions) | Adam Murray |
| April 28 | 2.1 (Galilean algebra) | Govert Nijs |
| May 5 6 | 6.1 (Noether charges are gauge invariant) | Peter Kristel |
| May 12 | 3.1 (Young diagrams of the wrong shape) | Ruben Doornenbal |
| May 12 | 11.1 (supersymmetry for N=1 vector) | Konstantina Polydorou |
| May 23 22 | 10.1 (anti/symmetry of gamma’s) | Peter Kristel |
| May 28 | 13.1 (framelike analogue of Fronsdal action) | Konstantina Polydorou |
| May 28 | 14.1 (3d massless spinor field unfolded) | Adam Murray |
| May 28 | 14.2 (3d conformal algebra via y’s) | Bahman Najian |
Exam and retake
Exam: There is a written exam in the afternoon of May 26, at 13:30–16:30 in MIN 202.
Retake: There is a retake in the afternoon of July 7, at 14:00–17:00 in MIN 401 204.
Rules: Students are allowed to bring a copy of the lecture notes, which may contain handwritten notes, as well as their own solutions from the tutorials.
Material: The exam covers all material from the lecture notes up to and including Section 15.3, but
- Footnotes and starred problems may be skipped;
- Sections 3.4 (self-duality) and 11.4.1 (massive supermultiplets) may be skipped;
- In Section 13.3 the proof of the first on-shell theorem (including σ_ cohomology) may be skipped, although the result (13.34) and what that means are part of the exam;
- In Section 14.3 the (hints for the) proof of the Campbell-Hausdorff formula may be skipped. The same holds for the discussion of P-Q (Wick) ordering at the end of that section.
Further information for students
This course qualifies as a primary optional course (7.5 ECTS) in theoretical physics. Please register through Osiris. Note that familiarity with quantum field theory is assumed.
Examination: the final grade will consist of student presentations of starred problems (at most two presentations per student, each contributing up to one bonus point) and a written exam on May 26. The date of the retake will be decided together with the students once the grades for the exam are known.
More information: for further questions you may contact Jules (j.lamers”at”uu.nl, room 410) or Professor Vasiliev (vasiliev”at”td.lpi.ru, room 422).