Abstract for Lecture III:
I will start with an introduction to topological quantum field theory (TQFT), and then explain the current theoretical understanding of the fractional quantum Hall effect based on TQFT’s. Finally I will outline my current research on the classification of TQFT’s. The central focus is a conjecture that I made in 2003: if the number of simple particle types in a TQFT is fixed, then there are only finitely many TQFT’s.
TQFT was invented by Witten to explain the Jones polynomial. It was used by condensed matter physicists in the 1980’s to describe the fractional quantum Hall effect. This completely unexpected effect was discovered by Tsui and Stormer in 1982 in a 2-dimensional electron gas under extreme physical conditions. For this work, they were awarded the Nobel prize in 1998.
The emerging theme is the existence of new states of matter: topological states matter. These new states of matter can be described by TQFT’s. They are best regarded as a kind of quantum crystal. Pursuing this analogy, we may ask for their classification. Classical crystals are classified by pointed space groups. Quantum crystals should be classified by certain “quantum groups”. The notion that is useful is Turaev’s modular tensor category. Therefore, we are asking about the classification of modular tensor categories.
Electrons in flatland continue to dazzle physicists and mathematicians with surprising phenomena. The nexus between quantum topology, quantum physics and quantum computing is fascinating, and will lead us to many hidden treasures.
Reference for Lecture
3:
Axioms for topological quantum field theories (V. Turaev). Annales de la faculté des sciences de Toulouse Sér. 6, 3 no. 1 (1994), p. 135-152