Maths projects → Project IV topics
Project IV 2018-2019
Supersymmetry and Seiberg duality
Description
Some properties of Nature seem at first to defy understanding, so it is astonishing when they finally yield to a mathematical treatment. One example is related to the strong force that binds nuclei together. This is a so-called strongly coupled field theory, which is another way of saying that if you try to pull a proton apart, the constituent quarks simply produce more quarks and stick to those instead. Ultimately the energy you provide just produces more and more lumps similar to the original proton. At large distances and low energies the quarks and the glue binding them are just a gigantic sticky mess which is on the face of it very difficult to describe mathematically. On the other hand at high energies and short distances we can describe the interactions between quarks very well (by essentially ignoring the effect of the glue); the theory that does this is known as Quantum Chromodynamics (QCD). One of the most interesting developments in recent years was the discovery by Seiberg of a way to describe the long range physics in terms of much simpler theories similar to the original QCD. Seiberg essentially derives the lumps, i.e. the protons and mesons, and the interactions between them, that describe the long range physics. Unfortunately the theories in which one can do this are not quite real-world QCD but are supersymmetric versions of it: so far it has proven impossible to find a dual for real-world QCD.
This challenging project will aim to develop an understanding of Seiberg duality. As it operates in supersymmetric field theory, you will first need to be or to become familiar with Quantum Mechanics, Quantum Field Theory, fermions and the Dirac equation, the renormalization group and asymptotic freedom, and Supersymmetry. Then you will learn about the properties and tests of Seiberg duality. From there you could learn about the so-called duality cascade and the links to higher dimensional gravity and string theory. Alternatively (or perhaps as well!) you would be able to consider applications to particle physics.
Prerequisites
Analysis in many variables II, Mathematical Physics II, Quantum Mechanics III and a co-requisite is Advanced Quantum Theory IV
Resources and references
There is a large number of papers and some books discussing the mathematics;
A good overviews of quantum mechanics is contained in Hey and Walters and also in The Feynman lectures on physics / Feynman, Leighton, Sands. Then try the introduction in Quantum mechanics / Leonard I. Schiff. Also the QMIII course notes.
A basic familiarity with quantum field theory will be required. Introductory texts on quantum field theory such as Quantum Field Theory in a Nutshell by A.Zee, Princeton University Press, Quantum Field Theory by Itzykson, C. and Zuber, J. B. New York: McGraw-Hill, 1980. Another good introduction to quantum field theory is Aitchison and Hey
A basic discussion of Supersymmetry is Srivastava Supersymmetry superfields and supergravity and the classic book by Supersymmetry and Supergravity by Wess and Bagger, Princeton University Press.
A great overview of the properties of supersymmetric field theory and Seiberg duality can be found in http://arXiv.org/abs/hep-th/9509066 and also in Modern Supersymmetry by Terning OUP.
An excellent review of Seiberg duality and the duality cascade can be found in Strassler's review, http://arXiv.org/abs/hep-th/0505153