The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory

Abstract

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.

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Cited by in corpus (7)

• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• A Geometric Quantisation view on the AJ-conjecture for the Teichmüller TQFT
• Genus-one complex quantum Chern--Simons theory
• Level N Teichmüller TQFT and Complex Chern-Simons Theory
• Asymptotic properties of the Hitchin-Witten connection
• Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties
• Asymptotic aspects of the Teichmüller TQFT