Aarhus

The AJ conjecture relates two different knot invariants, namely the coloured Jones polynomial and the A-polynomial. The approach we will use will be that of 2+1 dimensional Topological Quantum Field Theory. Indeed, the coloured Jones polynomial is constructed in Reshetikhin and Turaev's formulation of a TQFT using quantum groups. The A-polynomial is defined by a subvariety of the moduli space of flat SL(2,C) connections of a torus.  Geometric quantization on this moduli space also gives a TQFT, and the correspondence between these provides a framework where the knot invariants can be compared. In the talk I will sketch the above constructions and show how we can do explicit calculations for simple knots. This is work in progress joint with J. E. Andersen.

I will briefly describe a differential geometric construction of Hitchin's projectively flat connection in the Verlinde bundle, over Teichm\"uller space, formed by the Hilbert spaces arising from geometric quantization of the moduli space of flat connections on a Riemann surface. We will work on a general symplectic manifold sharing certain properties with the moduli space. Toeplitz operators enter the picture when quantizing classical observables, but they are also closely connected with the notion of deformation quantization. Furthermore, through an intimate relationship between Toeplitz operators, the Hitchin connection manifests itself in the world of deformation quantization as a connection on formal functions. As we shall see, this formal Hitchin connection can be used to construct a deformation quantization, which is independent of the Kähler polarization used for quantization. In the presence of a symmetry group, this deformation quantization can (under certain cohomological conditions) be constructed invariantly. The talk presents joint work with J. E. Andersen.