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Centre for Particle Theory

Non-Perturbative Physics

16 Lectures Dr M. del Zotto

This course is meant to be an introduction to some non-perturbative aspects of quantum field theory.

The main focus of this course are solitons, topologically non-trivial finite-energy solutions of classical field theories that are non-perturbative in nature and nevertheless play crucial roles in various physical applications. Construction, classification and properties of various types of solitons for various theories in different space-time dimensions are considered.

Another focus of this course are anomalies. Besides being a cornerstone for the quan- tum consistency of gauge theories, anomalies have a plethora of other old and new non- perturbative applications.
Time permitting, we will give a brief introduction to the concept of duality, one of the most important developments in modern quantum field theory, which is intimately related to the main topics of the course.

Outline of the course

Solitons basics. Topology, Derrick's theorem, Bogomolny bounds, examples of solitons eg. kinks, vortices, lumps, Skyrmions.

Non-perturbative aspects of gauge theory.

  • Monopoles: Dirac monopoles, charge quantization, non-abelian monopoles, spontaneous symmetry breaking, BPS monopoles, moduli space metrics.
  • Instantons: Chern number, Self-dual Yang-Mills, 't Hooft ansatz, ADHM construction (if time).
  • Anomalies. Local and global anomalies, anomaly matching.

Introduction to dualities (if time permits). Thirring/sine-gordon and bosoniza- tion, Montonen-Olive conjecture, Seiberg duality in 3d.

Books for the course

Several online resources are made available to the students during the lectures, suitably tuned to the level of the class. Some good textbooks that covers most of the material of the course are:

N.S. Manton and P.M. Sutcliffe, Topological solitons (CUP, 2004)
R. Rajaraman, Solitons and instantons, An introduction to solitons and instantons in quantum field theory (North-Holland, 1982)
S. Coleman, Aspects of Symmetry (CUP, 1985)
Y. Shnir, Magnetic monopoles (Springer, 2005)
S. Weinberg, The Quantum Theory of Fields II, (Cambridge University Press, 1996)
C. Itzykson and J.-B. Zuber, Quantum Field Theory, (McGraw-Hill, 1980)