Subfactors, bimodules and planar algebras

Vanderbilt University, Nashville TN

Investigators

Abstract

This project on subfactors and planar algebras aims at developing a wide range of techniques to better understand the structure of subfactors and discover new examples. One of the main goals is to understand the technique of composing subfactors in terms of their associated planar algebras. This will lead to new methods of constructing subfactors and planar algebras, in particular new examples with infinite principal graphs which are essential for progress in the structure theory. Moreover, every such subfactor will provide a potentially very interesting fusion category of bimodules, which can be computed explicitly. As a first step, those compositions will be studied that arise as planar subalgebras of the tensor product of two Temperley-Lieb planar algebras. For more general compositions, a cohomology theory for subfactors and planar algebras has to be developed which will replace group cohomology. A very rich theory is likely to emerge from these investigations. In particular, it is expected that new classification results for subfactors with Jones index two times the golden ratio squared will be obtained. A subfactor can be viewed as a mathematical object which captures quantum symmetries of a mathematical or quantum physical system. These symmetries play a key role in understanding the behavior of these complex systems, and the theory of subfactors provides effective tools to manipulate and study them. This theory has had many profound and surprising applications to numerous areas of mathematics and physics, such as conformal field theory, statistical mechanics, low dimensional topology and combinatorics. Subfactors have contributed in an important way to the understanding of naturally occurring structures in these a priori quite distinct areas of mathematics and physics. It is expected that the project will make important contributions to some of these areas of basic science. Exciting applications of planar algebras and their associated fusion categories to solid state physics and to topological quantum computing are possible. Furthermore, the project will involve graduate students and postdoctoral researchers and contribute to their training as researchers in mathematics.