We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in K\"{ a}hler quantisation, with polarisations coming from the nonabelian Hodge hyper-K\"{ a}hler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.
Categories: math.QA math-ph math.MP