2014/09/03 by Jørgen Ellegaard Andersen, Andersen, Jørgen Ellegaard, Niels Leth Gammelgaard +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.DG #math.QA
paper · pdf · doi:10.48550/arxiv.1409.1035
arxiv created 2014/09/03 · openalex publication_date 2014/09/03 · arxiv updated 2014/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler structures do not admit holomorphic vector fields. Following Witten, we define a complex variant of the Hitchin connection on the bundle of prequantum spaces. The curvature is essentially unchanged, so projective flatness holds in the same cases. Finally, the results are applied to quantum Chern-Simons theory, both for compact and complex gauge groups.