Wahlpflichtvorlesung für So.Se. 2013:

“Topological Objects in (Quantum) Field Theory”

von Priv. Doz. Dr. Francesco Giacosa

Zeit (Time) (2 Std) Fr, 12:00 (c.t.)-14:00, wöchentlich (weekly).

Erster Termin (first lecture): 19/4/2013.

Raum (Room): 2.114 (Aquarium-Raum)

Web-page: http://th.physik.uni-frankfurt.de/~giacosa/topobj.html

Email: giacosa@th.physik.uni-frankfurt.de

The goal of the lecture is to describe some examples of topological objects: these are highly interesting and non-trivial solutions of nonlinear field theories. Such solutions can be compared to huge and stable waves. Lower dimensional field theories represent a good starting point due to their simplicity and the possibility to visualize some topological stable configurations. The related concept of instantons is also introduced in this context. Its role for quantum mechanics is explained with simple examples.

Then, the attention is driven to four-dimensional field theories. In the framework of electrodynamics the monopole is described. The rich topological structure of Yang-Mills theory in four dimensions is addressed: the role of monopoles and instantons in Yang-Mills theory and, more in general, in QCD, is discussed.

Other interesting topological objects, such as the Skyrmion and the sphaleron, will be presented.

Themes

• Soliton in 1+1 dimensions: the kink. The concept of the topological charge.
• The concept of instanton: simple example in Quantum Mechanics
• Derrick’s theorem.
• Solitons of the the O(3) model in 1+2 dimensions and instantons in 1+1 dimensions.
• The vortex.
• Magnetic monopole in Electrodynamics.
• Monopole and instantons in Yang-Mills-Theory.
• Topology and hadron physics (Skyrmion).
• Sphaleron.

Book on topology: R. Rajamran, Solitons and Instatnons

Lecture on topology: G. ‘t Hooft, Monopoles, Instantons, and confinement”. http://arxiv.org/pdf/hep-th/0010225v1.pdf

Handsout

- Introduction of the course (Fr. 19/4): pdf.

- Field theory in 1+1 dim, wave-packet spreading, the kink as a easy soliton (Fr. 26/4): pdf.

- The kink, part 2. Energy, Topological charge and current (Fr. 10/5): pdf. (Links: http://en.wikipedia.org/wiki/Soliton ; Video: https://www.youtube.com/watch?v=wEbYELtGZwI&list=PLKTMChjaq_EMzMJ3lIrTvrlT2fI1Rou4E&index=1)

- Sine-Gordon model. Derrik’s theorem (Fr. 17/5): pdf. (Link with animations of the sine-Gordon mode lhttp://en.wikipedia.org/wiki/Sine%E2%80%93Gordon_equation)

- Sine-Gordon (part II, 24/5/2013): scattering and bound state: pdf. Instanton in 1+0 dimensions and discussion of solitons/instantons: pdf.

- Soliton/instanton (part II, 31/5/2013): pdf. Solitions in the O(3) model in 1+2 dimensions: pdf.

- The nonlinear O(3) model in 1+1 dimensions (14/6/2013): pdf. The vortex solution in 1+2 dimensions with a U(1) photon field: pdf.

- The Aharonov-Bohm effect (21/6/2013): pdf. The Dirac magnetic monopole: pdf.

- The Dirac magnetic monopole (part 2, 5/7/2013): pdf. The dual-superconductivity picture of confinement: pdf

- ‘t Hofft-Polyakov Monopole in the Georgi-Glashow model and the dyon in SU(2) Yang-Mills (12/7/2013): pdf.

- Recall of the SU(N) Yang-Mills theory and the phenomenon of dimensional transmutation (12/7/2013): pdf.

- Instatons in SU(2)-Yang-Mills and their role in QCD (19/7/2013): pdf.

Next lectures with student’s presentations:

Wednesday 21/8/2013 at 14:00 in room 2.114. (Subject of my presentation-lecture: the Polyakov loop: pdf).

Wednesday 18/9/2013 at 14:00 in room 2.114. (Subject of my presentation-lecture: the Skyrmion).

Remember to send me an Email with the choice of your chosen topic. First-come first serve...

Contact: giacosa@th.physik.uni-frankfurt.de

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