Description

Supersymmetric field theories are a particular class of quantum field theories that enjoy a surprising form of symmetry known as supersymmetry, which relates bosonic fields (like the force carriers, such as photons) to fermionic fields (which provide the basic matter particles in Nature, such as electrons and quarks).

The presence of supersymmetry in a theory makes it easier to study. One of the surprises that arose from the study of supersymmetric theories in the 80s and 90s is that they often enjoy the phenomenon of duality: a single quantum field theory admits two (or more) very different semi-classical approximations. Very often, there are interesting questions that are very hard to answer in one classical description that become easier to solve in the dual description. As it turns out, duality is pervasive in supersymmetric quantum field theory and in string theory.

In the first term we will study the basics of some important instances of duality

• Electromagnetic duality in electromagnetism: why do Maxwell’s equations in the absence of charge and currents look the same for electric and magnetic fields?
• T-duality: why the strings in string theory cannot tell apart very small circles from very large ones.
• Universality: why different Quantum Field Theories tend to look very similar if your experiments do not involve very high energies.
• Holography: why some quantum field theories in four dimensions are secretly the same as string theories living in ten dimensions.

During the second term you could choose to go deeper into any of the topics we explored in the first term, or could discuss some other topics related to dualities:

• More generally, are there cases of \(n\)-ality with \(n>2\)? That is, are there quantum theories with \(n>2\) different classical limits?
• What about $n=0$? That is, are there quantum theories with no classical Lagrangian description? How do we even define these theories?
• How do you test duality?
• In the 80s string theorists discovered 5 different versions of supersymmetric field theory in ten dimensions. In the 90s, it was understood that they are all likely to be secretly the same theory, a mysterious (and still poorly understood) quantum object sometimes known as M-theory. How can this be? What do we know about M-theory?

Mode of operation and evidence of learning

How will it operate? This is an individual project based on guided independent study. The project will revolve around learning through reading with focus on the underlying theory, and the development of deep understanding.

How will students demonstrate understanding? Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Pre-requisites and co-requisites

Reading material

• Dualities of Fields and Strings by Joe Polchinski.
• Duality in Quantum Field Theory (and String Theory) by Luis Alvarez-Gaumé and Frederic Zamora.