Subfactors and their connections with low dimensional topology, and low dimensional physics

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The "subfactors" in the title arise in quantum physics as part of the mathematical machinery needed to describe the universe compatibly with the requirement that information can travel no faster than the speed of light. Subfactors have numbers associated with them that can be used to parametrize possible physical theories. One such parameter is known as the "statistical dimension." It measures the complexity of what happens when particles in a system are changed amongst one another. If space-time is low dimensional this particle exchange is very subtle because particles have no room to go around each other. This causes the numbers associated to subfactors to be useful, as well, in the description of space-time itself. Such ideas are thought to provide possible architectures for quantum computing.The project will focus on the following three main areas, all related to subfactors:(1) Advancing research in planar algebras. Bigelow's jellyfish technique is a close cousin of the Yang-Baxter equation. Generaliztions of this technique should allow one to push the classification of low-index subfactors to index 6 and possibly beyond.(2) Low-dimensional quantum field theory. Wassermann showed that the fusion operation in WZW conformal field theory can be interpreted as the Connes fusion of bimodules over algebras of local observables. This work will be extended most importantly by establishing the existence of a manifestly unitary version of Segal conformal field theory. (3) Continuum limits for arbitrary planar algebras. The principal investigator will study the question: Do all subfactors come from conformal field theories? He proposes a first naive approach to this question using the planar algebra picture. Letting the boundary points on the circle become dense is the continuum limit process in statistical mechanics.