Aspects of Quantum Mathematics Hitchin Connections and AJ Conjectures
We discuss two different areas related to $(2+1)$-dimensional Topological Quantum Field Theory, namely geometric quantization and the AJ-conjecture in knot theory. First, we construct a Hitchin connection in geometric quantization with metaplectic correction of symplectic manifolds, and compare it to previous constructions. Second, we review the AJ-conjecture in knot theory, relating the coloured Jones polynomial and the A-polynomial. We reformulate this conjecture geometrically, drawing on geometric quantization of moduli spaces. Last, we use Faddeev's quantum dilogarithm to describe the asymptotic behaviour of the coloured Jones polynomial and show how to obtain the A-polynomial from this for twist knots.
Thesis advisors: Jørgen Ellegaard Andersen and Nicolai Reshetikhin