The Topology of Quantum Invariants

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DMS-0103922Justin D. Roberts The goal of this project is to try to understand the topology underlying quantum invariants in three dimensions. The current state of knowledge seems inadequate for explaining what kind of topological information the invariants carry, and therefore what kind of applications in three-dimensional topology they might have. However ithas revealed (through the frameworks of TQFT and Vassiliev theory) a wealth of very interesting algebraic structures, which seem to be specifically related to topology in three dimensions, and so should be thought of as kind of ``new algebraic topology'' in three dimensions.Roberts intends to try to bridge the gap between this new topology and classical algebraic topology. There are three main areas of investigation. The first is K-theory, which could help to explain the relationship between quantum knot invariants and Vassiliev invariants (for example, why the Jones polynomial is a polynomial). The second is Rozansky-Witten theory, a very interesting new example of a TQFT which provides new hints about intepretation of the invariants, and could have additional applications in complex or hyperkaehler geometry. The third is the hyperbolic volume conjecture of Kashaev, Murakami and Murakami. This conjecture seems to fit into a general pattern of ideas relating quantization (and specifically, 6j-symbols) to classicalgeometry; ideas which might provide a conceptual context for the conjecture.To a mathematician, a knot is what you get by tangling up a piece of string and then gluing its ends together. It is clear that there are qualitatively different ways of doing this; trying to describe and understand these ways forms a branch of topology, which might be thought of as what remains of geometry when quantitative questions are ignored. The main tools of knot theory are "topological invariants", numbers which can be associated to knots and which encode someinformation about their structure. During the last sixteen years, ideas borrowed from the physics of quantum field theory have led to the discovery of many new and beautiful invariants called "quantum invariants", but because of this unusual origin, their meaning remains very mysterious, and their potential unfulfilled. The goal of the project is to use some new ideas to try to explain this mystery and tobridge some of the gaps between geometry, topology, and physics. The work should contribute to the healthy exchange of ideas between these disciplines, and stimulate new work in each.