Genus-one complex quantum Chern--Simons theory

Abstract

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 Kähler quantisation, with polarisations coming from the nonabelian Hodge hyper-Kähler 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.

arXiv commentVersion 3, 25 pages (plus references), comments welcome

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Cited by in corpus (2)

• A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT
• $\mathrm{Sp}(1)$-symmetric hyperkähler quantisation