Genus-one complex quantum Chern-Simons theory (Q6042872)

The aim of the paper is to unify two approaches to the geometric quantization of complex Chern-Simons theory for genus one closed surfaces, by considering the natural complexified analogue of the Hitchin connection (previously used for the structure group \(\mathrm{SU}(n)\) and relating it to the Hitchin-Witten connection (used for \(\mathrm{SL}(n, \mathbb{C})\) via the Bargmann transformation. The main four results are as follows: Theorem 1. The complexified Hitchin connection preserves holomorphicity. Moreover, it is flat and mapping class group invariant. Theorem 2. The Bargmann transform interwines the lifted Hitchin-Witten and complexified Hitchin connections when acting on (sufficiently regular) families of polarised sections, in a mapping class group equivariant fashion. Theorem 3. For every value of the Teichmüller parameter, the Bargmann transform defines a unitary isomorphism between the corresponding Hilbert spaces arising from geometric quantisation on the moduli space. Theorem 4. The Bargmann transform on the moduli spaces interwines the Hitchin-Witten and complexified Hitchin connections.