Subfactors and Symmetry

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Abstract Bisch The main goal of this project is to gain a deeper understanding of the structure of subfactors and their standard invariants. The novel planar algebra techniques will be used to investigate and construct subfactors through a generators and relations approach to the associated planar algebras. This will build on prior work of Bisch and Jones in which a complete classification of singly generated planar algebras subject to a certain natural dimension condition was given. For instance, the Fuss-Catalan algebras of Bisch and Jones, whose representations have been used to construct new integrable lattice models in statistical mechanics, appear as an important part of this classification program. Annular Fuss-Catalan algebras will be investigated with the ultimate goal of using them to answer questions related to intermediate subfactors. Obstructions for compositions of subfactors and rigidity properties of subfactors will be analyzed. Potential applications of subfactor ideas and techniques to statistical mechanics and quantum information theory will be investigated. Ideas from operator algebras and non-commutative geometry have gained increasing importance in conformal field theory, string theory and quantum information theory. It has become clear that the symmetries of non-commutative spaces and quantum physical systems can no longer be understood through classical mathematical objects alone (such as groups). A subfactor can be viewed as a mathematical object that captures these new symmetries and operator algebra techniques can be used to analyze and understand them. A particular intriguing structure that appears naturally in this context are the planar algebras, a structure that allows one to compute with very abstract mathematical objects by simply manipulating planar diagrams topologically. Such a formalism is tailor-made for applications in statistical mechanics and new solvable models based on these ideas have already been constructed. One of the founders of the theory of operator algebras was John von Neumann who in the 1930's discovered that certain algebras of operators on a Hilbert space are the natural framework for understanding symmetries of quantum physical systems. His ideas play an important role in quantum mechanics and fundamental laws of nature such as the Heisenberg uncertainty principle appear as a natural consequence of von Neumann's abstract theory. The theory of subfactors, which is based on von Neumann's concepts, had an important impact across several fields in mathematics and physics with profound applications to knot theory, low dimensional topology, statistical mechanics and conformal field theory to mention just a few.