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.

We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kähler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.

Abstract. We give a differential geometric construction of a connection, which we call the Hitchin connection, in the bundle of quantum Hilbert spaces arising from metaplectically corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid family of Kähler structures, all of which give vanishing first Dolbeault cohomology groups.This generalizes work of both Hitchin, Scheinost and Schottenloher, and Andersen, since our construction does not need that the first Chern class is proportional to the class of the symplectic form, nor do we need compactness of the symplectic manifold in question.Furthermore, when we are in a setting similar to the moduli space, we give an explicit formula and show that this connection agrees with previous constructions.Mathematics Subject Classification (2010). 53D50, 32Q55.

a b s t r a c tLet X be a compact connected Riemann surface of genus at least two. Let M H (r, d) denote the moduli space of semistable Higgs bundles on X of rank r and degree d. We prove that the compact complex Bohr-Sommerfeld Lagrangians of M H (r, d) are precisely the irreducible components of the nilpotent cone in M H (r, d). This generalizes to Higgs G-bundles and also to the parabolic Higgs bundles.

No abstract