Journal of Symplectic Geometry

Volume 20 (2022)

Number 6

Genus-one complex quantum Chern–Simons theory

Pages: 1215 – 1253



Jørgen Ellegaard Andersen (Centre for Quantum Geometry (QM), Danish Institute for Advanced Study, Syddansk Universitet (SDU), Copenhagen, Denmark)

Alessandro Malusà (Department of Mathematics, University of Toronto, Ontario, Canada)

Gabriele Rembado (Hausdorff Centre for Mathematics, Universität Bonn, Germany)


We consider the geometric quantisation of Chern–Simons theory for closed genus-one surfaces and semisimple complex algebraic 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.

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Received 20 March 2021

Accepted 23 June 2022

Published 26 April 2023