  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.
