Vanishing Quantum Dimensions in Low-dimensional Topology

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The first main theme of the proposed research is quantum 6j-symbols and 3-manifold invariants. The Principal Investigator, Patureau-Mirand and Turaev have shown that categories with vanishing quantum dimensions naturally give rise to modified 6j-symbols and re-normalized link and 3-manifold invariants. The project continues this work with a goal of further understanding these objects by deriving explicit formulas for these modified 6j-symbols and constructing Topological Quantum Field Theories (TQFTs) from the re-normalized 3-manifold invariants. This work is related to the Volume Conjecture. The proposed research will also focus on constructing generalized Kashaev-type quantum hyperbolic invariants of 3-manifolds with an aim of defining Chern-Simons-type invariants for non-compact groups. A final theme of the proposed research is the construction of new state-sum invariants of shadows. With the influx of quantum field theory into low-dimensional topology, the past two decades have seen the emergence of a revolutionary point of view in the theory of link and 3-manifold invariants. The discovery of the V. Jones polynomial by Jones in 1984 and its 3-dimensional quantum field theory interpretation by E. Witten in 1989 have opened the door for the use of new algebraic techniques to study topology. These developments led to a new branch of mathematic known as "quantum topology." Many useful and interesting topics of quantum topology are enriched by vanishing quantum dimensions; for example the construction of quantum 3-manifold invariants and the Volume Conjecture both depend on the subtleties of such vanishing. This project has three components, all concerning the vanishing of quantum dimensions in low-dimensional topology.